A fermionic representation is given for all the quantities entering in the generating
function approach to weighted Hurwitz numbers and topological recursion. This
includes:KPand 2D Toda τ -functions of hypergeometric type, which serve as generating
functions for weighted single and double Hurwitz numbers; the Baker function, which
is expanded in an adapted basis obtained by applying the same dressing transformation
to all vacuum basis elements; the multipair correlators and the multicurrent correlators.
Multiplicative recursion relations and a linear differential system are deduced for the
adapted bases and their duals, and a Christoffel–Darboux type formula is derived for
the pair correlator. The quantum and classical spectral curves linking this theory with
the topological recursion program are derived, as well as the generalized cut-and-join
equations. The results are detailed for four special cases: the simple single and double
Hurwitz numbers, the weakly monotone case, corresponding to signed enumeration of
coverings, the strongly monotone case, corresponding to Belyi curves and the simplest
version of quantum weighted Hurwitz numbers.